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

<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">aC</font> =  <a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">zC</font> =  <a href="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">lC</font> =  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">AA</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> #);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<a NAME="E1:69_1_1"/><span class="lab"><font color="Green" title="E50">A0</font></span>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#D11" target="_self" title="LPSPACE2:def.11">VSPDef6X</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:69_1_1"/>
 <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A1</font>, <font color="Maroon" title="c12">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</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="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A1</font></span><br/>

<a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E4:69_1_1_1"/>
<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:69_1_1_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
( <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E7:69_1_1_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E6:69_1_1_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/>
<a NAME="E8:69_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:69_1_1_1"/>
<font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E7:69_1_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</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">thus </span><a NAME="E3:69_1_1"/>
 <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A1</font>, <font color="Maroon" title="c12">A2</font>, <font color="Maroon" title="c13">A3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font></span>)</span></span><br/>

<a NAME="E1:69_1_1_2"/>
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_2"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E4:69_1_1_2"/>
<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:69_1_1_2"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
( <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E7:69_1_1_2"/>
<font color="Maroon" title="c13">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">c</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E9:69_1_1_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
( <font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c16">c</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E10:69_1_1_2"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A2</font> &amp; <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E6:69_1_1_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E9:69_1_1_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/>
<a NAME="E11:69_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">A1</font>,<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c15">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">A2</font>,<font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a></span>;<br/>
<a NAME="E12:69_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A15</font></span>: 
( <font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font> &amp; <font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="ref" href="lpspace2.html#T25" target="_self" title="LPSPACE2:th.25">Th01aLp</a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E6:69_1_1_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E9:69_1_1_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c18">b1</font> = <font color="Maroon" title="c15">b</font>, <font color="Maroon" title="c19">c1</font> = <font color="Maroon" title="c16">c</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E6:69_1_1_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E9:69_1_1_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a></span>;<br/>
<a NAME="E14:69_1_1_2"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
( <font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">b1</font> &amp; <font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">c1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/>
<a NAME="E15:69_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/>
<a NAME="E16:69_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E17:69_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:69_1_1_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/>
<a NAME="E18:69_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:69_1_1_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E12:69_1_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:69_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:69_1_1_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E12:69_1_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></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">thus </span><a NAME="E4:69_1_1"/>
 <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</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="c11">A1</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="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font></span><br/>

<a NAME="E1:69_1_1_3"/>
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_3"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E4:69_1_1_3"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c12">a</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E6:69_1_1_3"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="ref" href="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">LmDef1Lp</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c16">z1</font> = <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="lpspace2.html#T23" target="_self" title="LPSPACE2:th.23">LmDef1Lp</a></span>;<br/>
<font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">z1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>
<br/>.= 
<font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T32" target="_self" title="LPSPACE2:th.32">Th09a</a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:69_1_1_3"/>
<font color="Maroon" title="c11">A1</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="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E3:69_1_1_3"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E4:69_1_1_3"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E6:69_1_1_3"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</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">thus </span><a NAME="E5:69_1_1"/>
 <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</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="c11">A1</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<a NAME="E1:69_1_1_4"/>
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_4"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/>
<a NAME="E4:69_1_1_4"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_4"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c12">a</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_4"><span class="lab"><font color="Green" title="E51">A26</font></span></a></span>;<br/>
<a NAME="E6:69_1_1_4"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_4"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="lpspace2.html#T26" target="_self" title="LPSPACE2:th.26">Th01bLp</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">A2</font> =  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>;<br/>
<a NAME="E7:69_1_1_4"/>
 <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_4"><span class="lab"><font color="Green" title="E53">A28</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">A2</font> =  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">A2</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</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="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">A2</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="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span></span><br/>

<a NAME="E9:69_1_1_4"/><span class="lab"><font color="Green" title="E54">A29</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="txt" href="#E3:69_1_1_4"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="lpspace2.html#T26" target="_self" title="LPSPACE2:th.26">Th01bLp</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E11:69_1_1_4"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">a1</font></span>)</span> &amp; <font color="Maroon" title="c17">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="lpspace1.html#K8" title="LPSPACE1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c16">v</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T31" target="_self" title="LPSPACE2:th.31">ThB11Z</a></span>;<br/>
<a NAME="E12:69_1_1_4"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a></span>;<br/>
<a NAME="E13:69_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a>, <a class="txt" href="#E11:69_1_1_4"><span class="lab"><font color="Green" title="E55">A30</font></span></a></span>;<br/>
<a NAME="E14:69_1_1_4"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T42" target="_self" title="LPSPACE2:th.42">EQC02bx</a>, <a class="txt" href="#E11:69_1_1_4"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:69_1_1_4"/>
<font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c15">A2</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="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_4"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="txt" href="#E9:69_1_1_4"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></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>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x0</font>, <font color="Maroon" title="c12">y0</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> holds <br/> ( <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">A1</font>, <font color="Maroon" title="c14">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <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="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c11">x0</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c12">y0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E2:69_1_1_5"/>
<font color="Maroon" title="c13">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:69_1_1_5"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
( <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/><a NAME="E5:69_1_1_5"/>
<font color="Maroon" title="c14">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E7:69_1_1_5"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
( <font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/><a NAME="E8:69_1_1_5"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A1</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:69_1_1_5"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/><a NAME="E9:69_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">A1</font>,<font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a></span>;<br/><a NAME="E10:69_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A40</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="ref" href="lpspace2.html#T25" target="_self" title="LPSPACE2:th.25">Th01aLp</a>, <a class="txt" href="#E4:69_1_1_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:69_1_1_5"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">a1</font> = <font color="Maroon" title="c17">a</font>, <font color="Maroon" title="c20">b1</font> = <font color="Maroon" title="c18">b</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K2" title="LPSPACE2:func.2">RLSp_LpFunct</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:69_1_1_5"><span class="lab"><font color="Green" title="E52">A37</font></span></a></span>;<br/><a NAME="E12:69_1_1_5"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
( <font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c20">b1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font> &amp; 1 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a></span>;<br/><a NAME="E13:69_1_1_5"/>
<font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/><a NAME="E14:69_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c20">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a></span>;<br/><a NAME="E15:69_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c20">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>;<br/><a NAME="E16:69_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A45</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c18">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/><a NAME="E17:69_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></span>;<br/><a NAME="E18:69_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="lpspace2.html#T29" target="_self" title="LPSPACE2:th.29">ThB10</a></span>;<br/><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> = 
<font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c19">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
;<br/><a NAME="E20:69_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A48</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T30" target="_self" title="LPSPACE2:th.30">ThB11</a></span>;<br/><a NAME="E21:69_1_1_5"/>
( <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E8:69_1_1_5"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a></span>;<br/><a NAME="E22:69_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A49</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">A2</font> &amp; <font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a>, <a class="ref" href="lpspace2.html#T26" target="_self" title="LPSPACE2:th.26">Th01bLp</a>, <a class="txt" href="#E4:69_1_1_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E7:69_1_1_5"><span class="lab"><font color="Green" title="E52">A37</font></span></a></span>;<br/><a NAME="E23:69_1_1_5"/>
<font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c18">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E16:69_1_1_5"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E10:69_1_1_5"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a></span>;<br/><span class="kw">hence </span><a NAME="E24:69_1_1_5"/>
<font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E22:69_1_1_5"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</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">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font> )</span><br/><a NAME="E25:69_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E18:69_1_1_5"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="txt" href="#E8:69_1_1_5"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a></span>;<br/><span class="kw">hence </span><a NAME="E26:69_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:69_1_1_5"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="ref" href="lpspace2.html#D8" target="_self" title="LPSPACE2:def.8">VSPDef3X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></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">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font> )</span><br/><a NAME="E27:69_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a>, <a class="txt" href="#E20:69_1_1_5"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E8:69_1_1_5"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a></span>;<br/><span class="kw">hence </span><a NAME="E28:69_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y0</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:69_1_1_5"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font></span><br/><a NAME="E29:69_1_1_5"/>
1 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c17">a</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_1_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E30:69_1_1_5"/>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_5"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E8:69_1_1_5"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="lpspace2.html#D10" target="_self" title="LPSPACE2:def.10">VSPDef5X</a>, <a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E50">A0</font></span></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">hence </span><a NAME="E7:69_1_1"/>
(  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</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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