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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/>

<a NAME="E1:48"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  ex <font color="Olive" title="b6">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</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> <span class="p2">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="ref" href="clopban2.html#T19" target="_self" title="CLOPBAN2:th.19">Th19</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">RBLOP</font> =  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c3">BLOP</font> =  <a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c4">BS</font> =  <a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">ADD</font> =  <a href="csspace.html#K8" title="CSSPACE:func.8">Add_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">MULT</font> =  <a href="clopban2.html#K5" title="CLOPBAN2:func.5">FuncMult</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">ZE</font> =  <a href="csspace.html#K10" title="CSSPACE:func.10">Zero_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">EXMULT</font> =  <a href="csspace.html#K9" title="CSSPACE:func.9">Mult_</a> <span class="p1">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">UNIT</font> =  <a href="clopban2.html#K6" title="CLOPBAN2:func.6">FuncUnit</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c10">NRM</font> =  <a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:48"/><i><font color="Green" title="E28">A2</font></i>: 
( ( for <font color="Olive" title="b1">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> ) )
 
<b>by </b><i><a class="ref" href="clopban2.html#T19" target="_self" title="CLOPBAN2:th.19">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" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c13">a</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> ;<br/><i><font color="Green" title="E29">A3</font></i>: <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Add_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">y1</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
;<br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c14">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c15">y1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c15">y1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U2" title="CLVECT_1:sel.2">norm</a> of <span class="p2">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E4:48_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban2.html#E2:48_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="clvect_1.html#D10" title="CLVECT_1:def.10">CLVECT_1:def 10</a></i>;<br/><i><font color="Green" title="E30">A4</font></i>: <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c14">x1</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> = 
<span class="p1">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x1</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
;<br/><a NAME="E6:48_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="clopban1.html#K14" title="CLOPBAN1:func.14">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E7:48_1"/>
( <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 iff <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 <b>by </b><i><a class="txt" href="clopban2.html#E5:48_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="clvect_1.html#D10" title="CLVECT_1:def.10">CLVECT_1:def 10</a></i>;<br/><b>thus </b><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> = 
the <a href="clvect_1.html#U2" title="CLVECT_1:sel.2">norm</a> of <span class="p1">(<span class="default"><a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K9" title="CSSPACE:func.9">Mult_</a> <span class="p3">(<span class="default"><a href="clopban1.html#K8" title="CLOPBAN1:func.8">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban1.html#K13" title="CLOPBAN1:func.13">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D9" title="CLVECT_1:def.9">CLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c13">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="clopban2.html#E5:48_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="clvect_1.html#D10" title="CLVECT_1:def.10">CLVECT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
(  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">ComplexNormSpace-like</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="cfuncdom.html#V2" title="CFUNCDOM:attr.2">ComplexAlgebra-like</a> &amp;  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font> is <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">ComplexLinearSpace-like</a> )
 <b>by </b><i><a class="txt" href="clopban2.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="clopban2.html#E2:48"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="cfuncdom.html#D9" title="CFUNCDOM:def.9">CFUNCDOM:def 9</a>, <a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a>, <a class="ref" href="clvect_1.html#D10" title="CLVECT_1:def.10">CLVECT_1:def 10</a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>


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