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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/>



<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font></span>)</span> = 
<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </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/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </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="group_1.html#K1" title="GROUP_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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">z</font>, <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 
<b>by </b><i><a class="ref" href="lopban_4.html#D1" title="LOPBAN_4:def.1">LOPBAN_4:def 1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
 <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#T34" target="_self" title="CLOPBAN4:th.34">Th34</a></i>
<br/>.= 
 <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</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="clopban4.html#T36" target="_self" title="CLOPBAN4:th.36">Th36</a></i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z</font></span>)</span> = 
 <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2:48"><i><font color="Green" title="E44">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#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</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="clopban4.html#T36" target="_self" title="CLOPBAN4:th.36">Th36</a></i>
;<br/>


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