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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0 = 
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D2" target="_self" title="CLOPBAN4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">0 <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></i>
<br/>.= 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a>, <a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>

<a NAME="E2:29"/><i><font color="Green" title="E27">A1</font></i>: 
0 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="clopban4.html#K2" title="CLOPBAN4:func.2">Expan</a> 0,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8" title="SIN_COS:func.8">Coef</a> 0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D4" target="_self" title="CLOPBAN4:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2:29"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="sin_cos.html#T1" title="SIN_COS:th.1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10" title="SIN_COS:def.10">SIN_COS:def 10</a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/><br/>


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