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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/><b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<a NAME="E1:56"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 
<b>by </b><i><a class="ref" href="lopban_4.html#T40" target="_self" title="LOPBAN_4:th.40">Th40</a></i>;<br/>
<b>hence </b><i><font color="Green" title="E53">A2</font></i>: <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"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#T35" target="_self" title="LOPBAN_4:th.35">Th35</a></i>
<br/>.= 
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green" title="E54">A3</font></i>: <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"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> = 
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:56"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="lopban_4.html#T35" target="_self" title="LOPBAN_4:th.35">Th35</a></i>
<br/>.= 
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/>

<b>thus </b><a NAME="E4:56"/>
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E5:56"/>
 <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span>, <a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 <b>by </b><i><a class="txt" href="lopban_4.html#E2:56"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="lopban_4.html#E3:56"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="lopban_4.html#D1" target="_self" title="LOPBAN_4:def.1">Def1</a></i>;<br/>


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