<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="lopban_2.html#NM3">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/>



<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
;<br/>
<a NAME="E2:53"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font>, <a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="lopban_4.html#K10" 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="lopban_4.html#K10" 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="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i/>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="lopban_4.html#K10" 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="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z</font></span>)</span> = 
 <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i/>
;<br/>


</div>
