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<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">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<i><font color="Green">E48</font></i>:  <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z2</font></span>)</span> = 
 <a href="lopban_3.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z2</font></span>)</span> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="lopban_4.html#E2"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:50"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z1</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> <font color="Maroon">z2</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:50"><i><font color="Green">E47</font></i></a></i>
;<br/>
 <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</font></span>)</span> = 
 <a href="lopban_3.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</font></span>)</span> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="lopban_4.html#E2"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:50"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</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> <font color="Maroon">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:50"><i><font color="Green">E47</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:50"/>
(  <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z1</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> <font color="Maroon">z2</font></span>)</span> &amp;  <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</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> <font color="Maroon">z1</font></span>)</span> &amp;  <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lopban_4.html#K10" target="_self">exp</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z1</font></span>)</span> &amp;  <a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z1</font>, <a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</font> <a href="lopban_4.html#R1" target="_self">are_commutative</a>  )
 <b>by </b><i>, </i>;<br/>


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