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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">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/>



<a NAME="E1:54"/><i><font color="Green" title="E51">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">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="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</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="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="lopban_4.html#T38" target="_self" title="LOPBAN_4:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E2:54"/>
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c2">z</font> is <a href="lopban_3.html#V4" title="LOPBAN_3:attr.4">invertible</a>
 <b>by </b><i><a class="ref" href="lopban_3.html#D8" title="LOPBAN_3:def.8">LOPBAN_3:def 8</a></i>;<br/>

<b>hence </b><a NAME="E3:54"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="lopban_3.html#K9" title="LOPBAN_3:func.9">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_4.html#E1:54"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="lopban_3.html#D12" title="LOPBAN_3:def.12">LOPBAN_3:def 12</a></i>;<br/>

<b>thus </b><a NAME="E4:54"/>
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">z</font></span>)</span> is <a href="lopban_3.html#V4" title="LOPBAN_3:attr.4">invertible</a>
 <b>by </b><i><a class="txt" href="lopban_4.html#E1:54"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="lopban_3.html#D8" title="LOPBAN_3:def.8">LOPBAN_3:def 8</a></i>;<br/>

<b>hence </b><a NAME="E5:54"/>
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="lopban_3.html#K9" title="LOPBAN_3:func.9">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E1:54"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="lopban_3.html#D12" title="LOPBAN_3:def.12">LOPBAN_3:def 12</a></i>;<br/>


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