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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>; <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">n</font> being   <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> <br/> for <font color="Olive" title="b2">z</font>, <font color="Olive" title="b3">w</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="b2">z</font>,<font color="Olive" title="b3">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/>( <font color="Olive" title="b3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">w</font> &amp; <font color="Olive" title="b2">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font> )</span><br/><b>let </b><font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</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">z</font>,<font color="Olive" title="b2">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  holds <br/>( <font color="Olive" title="b2">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b1">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">z</font> )</span><br/><b>let </b><font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">w</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="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a>  implies ( <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> ) )</span><br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</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="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b>( <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> $1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> $1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> );<br/>
<i><font color="Green" title="E4">A2</font></i>: <font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></i>
;<br/>
<i><font color="Green" title="E5">A3</font></i>: <font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K6" title="CLOPBAN3:func.6">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="clopban3.html#D13" title="CLOPBAN3:def.13">CLOPBAN3:def 13</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#D12" title="CLOPBAN3:def.12">CLOPBAN3:def 12</a></i>
;<br/>
<i><font color="Green" title="E6">A4</font></i>: <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="txt" href="clopban4.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="txt" href="clopban4.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>
<font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="txt" href="clopban4.html#E3:3"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="txt" href="clopban4.html#E3:3"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
;<br/>
<a NAME="E6:3"/><b>then </b><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="clopban4.html#E4:3"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<a NAME="E7:3"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:3_5"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<i><font color="Green" title="E10">A8</font></i>: <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c4">w</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">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3_5"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="lopban_4.html#D1" title="LOPBAN_4:def.1">LOPBAN_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>
;<br/>
<font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c3">z</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="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3_5"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="lopban_4.html#D1" title="LOPBAN_4:def.1">LOPBAN_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:3_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="clopban4.html#E2:3_5"><i><font color="Green" title="E10">A8</font></i></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>
<a NAME="E8:3"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="clopban4.html#E6:3"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="clopban4.html#E7:3"><i><font color="Green" title="E8">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:3"/>
( <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="clopban3.html#K7" title="CLOPBAN3:func.7">#N</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
