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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">z1</font>, <font color="Maroon" title="c3">z2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ;<br/>

<i><font color="Green" title="E43">A2</font></i>:  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> = 
 <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:45"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="clopban4.html#E42"><i><font color="Green" title="E39">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z1</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> <font color="Maroon" title="c3">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
;<br/>
 <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">z1</font></span>)</span> = 
 <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:45"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="clopban4.html#E42"><i><font color="Green" title="E39">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</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> <font color="Maroon" title="c2">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E4:45"/>
(  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z1</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> <font color="Maroon" title="c3">z2</font></span>)</span> &amp;  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</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> <font color="Maroon" title="c2">z1</font></span>)</span> &amp;  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">z1</font></span>)</span> &amp;  <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c2">z1</font>, <a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  )
 <b>by </b><i><a class="txt" href="clopban4.html#E2:45"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="lopban_4.html#D1" title="LOPBAN_4:def.1">LOPBAN_4:def 1</a></i>;<br/>


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