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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/><font color="Olive" title="b1">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">z1</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</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="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font> )</span><br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E48">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</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="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font></span><br/>

<a NAME="E2:51"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2" title="LOPBAN_3:def.2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c2">z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D9" title="LOPBAN_3:def.9">LOPBAN_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#D2" target="_self" title="LOPBAN_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></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/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="lopban_4.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="lopban_4.html#D2" target="_self" title="LOPBAN_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c2">z1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> = 
<font color="Maroon" title="c2">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#D10" target="_self" title="LOPBAN_4:def.10">Def10</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="lopban_4.html#T6" target="_self" title="LOPBAN_4:th.6">Th6</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#T9" target="_self" title="LOPBAN_4:th.9">Th9</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K3" title="LOPBAN_3:func.3">*</a> <font color="Maroon" title="c2">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:51"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="lopban_4.html#T9" target="_self" title="LOPBAN_4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">rExpSeq</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="lopban_4.html#T7" target="_self" title="LOPBAN_4:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" title="LOPBAN_4:func.10">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="lopban_4.html#D10" target="_self" title="LOPBAN_4:def.10">Def10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
