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<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> = 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<a NAME="E3:2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:2_4"/>
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
;<br/><b>hence </b><a NAME="E3:2_4"/>
<font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:2"/>
( <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> )
 ;<br/>


</div>
