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





<b>thus </b><span class="p1">(<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 class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13">LOPBAN_3:def 13</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
;<br/>

<a NAME="E2:34"/><i><font color="Green">E47</font></i>: 
0 <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="lopban_4.html#K5" target="_self">Expan</a> 0,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K2" target="_self">Coef</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span>
<b>by </b><i>, , <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lopban_3.html#K10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="lopban_3.html#K11">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="lopban_3.html#K10">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13">LOPBAN_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13">LOPBAN_3:def 13</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
;<br/><br/>


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