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<b>let </b><font color="Maroon">X</font> be   <a href="lopban_2.html#NM3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>



<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1">LOPBAN_3:def 1</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
;<br/>
<b>hence </b><a NAME="E2:42"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 <b>by </b><i/>;<br/>


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