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

<span class="kw">set </span><font color="Maroon" title="c5">l</font> =  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font>;<br/>
<a NAME="E1:85_1_1"/>
 <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="85_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">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="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="clvect_1.html#D13" onmouseover="rs('clvect_1/D13')" onmouseout="rh()">CLVECT_1:def 13</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="c6">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E2:85_1_1_1"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E3:85_1_1_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D20" target="_self" onmouseover="rs('lpspacc1/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">aa</font> = <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>;<br/>
<a NAME="E4:85_1_1_1"/>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="lpspacc1.html#T41" target="_self" onmouseover="rs('lpspacc1/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<a NAME="E5:85_1_1_1"/><span class="kw">then </span>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T102" onmouseover="rs('mesfunc6/T102')" onmouseout="rh()">MESFUNC6:102</a></span>;<br/>
<a NAME="E6:85_1_1_1"/><span class="kw">then </span>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T25" onmouseover="rs('rfunct_1/T25')" onmouseout="rh()">RFUNCT_1:25</a></span>;<br/>
<a NAME="E7:85_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_1_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c8">a</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/>
<a NAME="E9:85_1_1_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
( <font color="Maroon" title="c9">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="lpspacc1.html#T41" target="_self" onmouseover="rs('lpspacc1/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E11:85_1_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c12">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">and </span><br/><a NAME="E12:85_1_1_1"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c12">g</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#D20" target="_self" onmouseover="rs('lpspacc1/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<a NAME="E13:85_1_1_1"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
( <font color="Maroon" title="c12">g</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c12">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:85_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="lpspacc1.html#T41" target="_self" onmouseover="rs('lpspacc1/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<a NAME="E14:85_1_1_1"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c12">g</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E11:85_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="lpspacc1.html#T45" target="_self" onmouseover="rs('lpspacc1/T45')" onmouseout="rh()">Th45</a>, <a class="ref" href="lpspacc1.html#T43" target="_self" onmouseover="rs('lpspacc1/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E15:85_1_1_1"/>
<font color="Maroon" title="c11">a1</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">a1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="lpspacc1.html#T45" target="_self" onmouseover="rs('lpspacc1/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E16:85_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:85_1_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="lpspacc1.html#T43" target="_self" onmouseover="rs('lpspacc1/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E17:85_1_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c9">f</font></span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c12">g</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_1_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E12:85_1_1_1"><span class="lab"><font color="Green" title="E57">A6</font></span></a></span>;<br/>
<a NAME="E18:85_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:85_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E14:85_1_1_1"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E13:85_1_1_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="lpspacc1.html#T47" target="_self" onmouseover="rs('lpspacc1/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:85_1_1_1"/>
( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:85_1_1_1"><span class="lab"><font color="Green" title="E60">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:85_1_1"/>
(  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspacc1.html#K24" title="LPSPACC1:func.24">L-1-CSpace</a> <font color="Maroon" title="c4">M</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace3.html#T2" onmouseover="rs('csspace3/T2')" onmouseout="rh()">CSSPACE3:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
