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<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_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/>



<a NAME="E1:51"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:51"/>
 <a href="clopban4.html#K8" target="_self">exp</a> <font color="Maroon">z</font> is <a href="lopban_3.html#V4">invertible</a>
 <b>by </b><i><a class="ref" href="lopban_3.html#D8">LOPBAN_3:def 8</a></i>;<br/>

<b>hence </b><a NAME="E3:51"/>
<span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <font color="Maroon">z</font></span>)</span> <a href="clopban3.html#K7">"</a>  <a href="hidden.html#R1">=</a>  <a href="clopban4.html#K8" target="_self">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="clopban3.html#D11">CLOPBAN3:def 11</a></i>;<br/>

<b>thus </b><a NAME="E4:51"/>
 <a href="clopban4.html#K8" target="_self">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> is <a href="lopban_3.html#V4">invertible</a>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="lopban_3.html#D8">LOPBAN_3:def 8</a></i>;<br/>

<b>hence </b><a NAME="E5:51"/>
<span class="p1">(<span class="default"><a href="clopban4.html#K8" target="_self">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="clopban3.html#K7">"</a>  <a href="hidden.html#R1">=</a>  <a href="clopban4.html#K8" target="_self">exp</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="clopban3.html#D11">CLOPBAN3:def 11</a></i>;<br/>


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