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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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>





<a NAME="E1:53"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font> <a href="clopban4.html#R1" title="CLOPBAN4:pred.1">are_commutative</a> 
 
<b>by </b><i><a class="ref" href="clopban4.html#T39" target="_self" title="CLOPBAN4:th.39">Th39</a></i>;<br/>
<b>thus </b><i><font color="Green" title="E50">A2</font></i>: <span class="p1">(<span class="default"><a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:53"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="clopban4.html#T34" target="_self" title="CLOPBAN4:th.34">Th34</a></i>
<br/>.= 
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>

<b>thus </b><i><font color="Green" title="E51">A3</font></i>: <span class="p1">(<span class="default"><a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:53"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="clopban4.html#T34" target="_self" title="CLOPBAN4:th.34">Th34</a></i>
<br/>.= 
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>

<b>thus </b><a NAME="E4:53"/>
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E5:53"/>
 <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span>, <a href="clopban4.html#K8" title="CLOPBAN4:func.8">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="clopban4.html#R1" title="CLOPBAN4:pred.1">are_commutative</a> 
 <b>by </b><i><a class="txt" href="clopban4.html#E2:53"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="clopban4.html#E3:53"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="clopban4.html#D1" target="_self" title="CLOPBAN4:def.1">Def1</a></i>;<br/>


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