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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/>

<a NAME="E1:48"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="complex1.html#K8">+</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="clopban2.html#T19" target="_self">Th19</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <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>set </b><font color="Maroon">c<sub>4</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>5</sub></font> =  <a href="csspace.html#K8">Add_</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>6</sub></font> =  <a href="clopban2.html#K5" target="_self">FuncMult</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="csspace.html#K10">Zero_</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="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>9</sub></font> =  <a href="clopban2.html#K6" target="_self">FuncUnit</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:48"/><i><font color="Green">E28</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K8">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K9">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="clopban2.html#T19" target="_self">Th19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="pre_topc.html#NM2">Point</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/><i><font color="Green">E29</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
the <a href="clvect_1.html#U2">norm</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K8">Add_</a> <span class="p3">(<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="p3">(<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> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U2">norm</a> of <span class="p2">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E4:48_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban2.html#E2:48_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><i><font color="Green">E30</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
;<br/> <a href="rlvect_1.html#K1">0.</a> <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="csspace.html#K10">Zero_</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>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E7:48_1"/>
( <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="clopban2.html#E5:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><b>thus </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
the <a href="clvect_1.html#U2">norm</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p3">(<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="p3">(<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="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13">BoundedLinearOperatorsNorm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="txt" href="clopban2.html#E5:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
(  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="clvect_1.html#V5">ComplexNormSpace-like</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cfuncdom.html#V2">ComplexAlgebra-like</a> &amp;  <a href="clopban2.html#K9" target="_self">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon">c<sub>1</sub></font> is <a href="clvect_1.html#V2">ComplexLinearSpace-like</a> )
 <b>by </b><i><a class="txt" href="clopban2.html#E1:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="clopban2.html#E2:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cfuncdom.html#D9">CFUNCDOM:def 9</a>, <a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>


</div>
