<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="struct_0.html#M1">Element</a> of  <a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>





<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="csspace.html#K9">Mult_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:17"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E2:17_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10">CLOPBAN1:def 10</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10">CLOPBAN1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8">CLOPBAN1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8">CLOPBAN1:def 8</a></i>;<br/>
<i><font color="Green">E16</font></i>: <span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
the <a href="clvect_1.html#U1">Mult</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/>
<a NAME="E7:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8">CLOPBAN1:def 8</a></i>;<br/>
<i><font color="Green">E17</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/>
<a NAME="E3:17_1_2"/>
 <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10">CLOPBAN1:def 10</a></i>;<br/>
<a NAME="E4:17_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban2.html#E2:17_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="clopban2.html#E2:17_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="clopban1.html#T40">CLOPBAN1:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> =  <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="clopban1.html#D8">CLOPBAN1:def 8</a></i>;<br/>
<i><font color="Green">E19</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/>
<a NAME="E7:17_1_2"/>
 <a href="clopban1.html#K11">modetrans</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="clopban1.html#D10">CLOPBAN1:def 10</a></i>;<br/>
<a NAME="E8:17_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban2.html#E3:17_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="clopban2.html#E6:17_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="clopban1.html#T40">CLOPBAN1:40</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clopban2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban2.html#E4:17_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="clopban2.html#E8:17_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="clopban1.html#D4">CLOPBAN1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clopban2.html#T4" target="_self">Th4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:17_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="clopban1.html#T40">CLOPBAN1:40</a></i>;<br/>
<b>hence </b><a NAME="E9:17_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8">BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="clopban2.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="clopban1.html#K11">modetrans</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="clopban2.html#E6:17_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="clopban2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban2.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</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="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clopban2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="clopban2.html#K4" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="clopban2.html#E1:17"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
