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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">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<i><font color="Green" title="E3">A1</font></i>: <font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/>
<font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K9" title="LOPBAN_3:func.9">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
;<br/>
<a NAME="E3:2"/><b>then </b><i><font color="Green" title="E4">A2</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="lopban_4.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E4:2"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_3"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<b>thus </b><font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K9" title="LOPBAN_3:func.9">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:2_3"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:2"/><i><font color="Green" title="E6">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="lopban_4.html#E3:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="lopban_4.html#E4:2"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><a NAME="E1:2_4"/>
<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="lopban_4.html#E5:2"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E2:2_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D13" title="LOPBAN_3:def.13">LOPBAN_3:def 13</a></i>;<br/><b>hence </b><a NAME="E3:2_4"/>
<font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E5:2"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2"/>
( <font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K10" title="LOPBAN_3:func.10">#N</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z</font> )
 ;<br/>


</div>
