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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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#V8" title="CLVECT_1:attr.8">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#V2" title="RLVECT_1:attr.2">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#V3" title="RLVECT_1:attr.3">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#V4" title="RLVECT_1:attr.4">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="algstr_0.html#V13" title="ALGSTR_0:attr.13">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#V3" title="GROUP_1:attr.3">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="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</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="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</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">vector-distributive</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#V3" title="CLVECT_1:attr.3">scalar-distributive</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#V4" title="CLVECT_1:attr.4">scalar-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">vector-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="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</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#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="clopban2.html#K9" title="CLOPBAN2:func.9">C_Normed_Algebra_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">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/>
<span class="kw">thus </span><a NAME="E1:45"/>
 <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#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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#V2" title="RLVECT_1:attr.2">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#V3" title="RLVECT_1:attr.3">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#V4" title="RLVECT_1:attr.4">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="algstr_0.html#V13" title="ALGSTR_0:attr.13">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#V3" title="GROUP_1:attr.3">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="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</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="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</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">vector-distributive</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#V3" title="CLVECT_1:attr.3">scalar-distributive</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#V4" title="CLVECT_1:attr.4">scalar-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">vector-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="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</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#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds <br/> ( (  not <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">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> ) &amp; (  not <font color="Maroon" title="c4">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> or <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">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> ) &amp; (  not <font color="Maroon" title="c4">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> or <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">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/>
<a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c7">x1</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 
;<br/>
<a NAME="E3:45_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/>
<span class="kw">hence </span><a NAME="E4:45_1"/>
( <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c4">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> )</span><br/>

<span class="kw">thus </span><span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>

<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span><br/>

<a NAME="E6:45_1"/>
( <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c7">x1</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E7:45_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c4">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/>
<a NAME="E2:45"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <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="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">thus </span><a NAME="E1:45_2"/>
<font color="Maroon" title="c5">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban2.html#T19" target="_self" title="CLOPBAN2:th.19">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:45"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
( ( 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="algstr_0.html#K1" title="ALGSTR_0: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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban2.html#T19" target="_self" title="CLOPBAN2:th.19">Th19</a></span>;<br/>
<a NAME="E4:45"/>
( ( 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="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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; <font color="Olive" title="b1">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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="struct_0.html#K5" title="STRUCT_0:func.5">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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0: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> ) ) &amp; ( 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="algstr_0.html#K1" title="ALGSTR_0: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="algstr_0.html#K1" title="ALGSTR_0: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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban2.html#T19" target="_self" title="CLOPBAN2:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45"/>
(  <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#V2" title="RLVECT_1:attr.2">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#V3" title="RLVECT_1:attr.3">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#V4" title="RLVECT_1:attr.4">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="algstr_0.html#V13" title="ALGSTR_0:attr.13">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#V3" title="GROUP_1:attr.3">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="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</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="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</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">vector-distributive</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#V3" title="CLVECT_1:attr.3">scalar-distributive</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#V4" title="CLVECT_1:attr.4">scalar-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">vector-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="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</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#V5" title="CLVECT_1:attr.5">scalar-unital</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E25">A3</font></span></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#D3" title="CLVECT_1:def.3">CLVECT_1:def 3</a>, <a class="ref" href="clvect_1.html#D4" title="CLVECT_1:def.4">CLVECT_1:def 4</a>, <a class="ref" href="clvect_1.html#D5" title="CLVECT_1:def.5">CLVECT_1:def 5</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="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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