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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="clopban4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> );<br/>
<i><font color="Green">E5</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
;<br/>
<i><font color="Green">E6</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
;<br/>
<i><font color="Green">E7</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> = 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E2:3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E2:3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<i><font color="Green">E8</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> = 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E3:3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E3:3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<a NAME="E6:3"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="clopban4.html#E4:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="clopban4.html#E5:3"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E7:3"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:3_5"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/>

<i><font color="Green">E12</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3_5"><i><font color="Green">E11</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="clopban4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
;<br/>
<i><font color="Green">E13</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3_5"><i><font color="Green">E11</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="clopban4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
;<br/>
<b>thus </b><a NAME="E4:3_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="clopban4.html#E2:3_5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="clopban4.html#E3:3_5"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="clopban4.html#E6:3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="clopban4.html#E7:3"><i><font color="Green">E10</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


</div>
