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

<b>set </b><font color="Maroon" title="c4">C</font> =  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c5">aC</font> =  <a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c6">zC</font> =  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c7">lC</font> =  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c8">A</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #);<br/>
<a NAME="E1:74_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">A1</font>, <font color="Maroon" title="c10">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #); <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">A1</font></span><br/>

<a NAME="E1:74_1_1_1"/>
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E3:74_1_1_1"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E4:74_1_1_1"/>
<font color="Maroon" title="c10">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E6:74_1_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
( <font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E7:74_1_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 
<b>by </b><i><a class="txt" href="#E3:74_1_1_1"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="#E6:74_1_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/>
<a NAME="E8:74_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c12">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>;<br/>
<b>hence </b><a NAME="E9:74_1_1_1"/>
<font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">A1</font>
 <b>by </b><i><a class="txt" href="#E7:74_1_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:74_1_1"/><i><font color="Green" title="E47">A6</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">A1</font>, <font color="Maroon" title="c10">A2</font>, <font color="Maroon" title="c11">A3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #); <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font></span>)</span></span><br/>

<a NAME="E1:74_1_1_2"/>
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><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> <b> such that </b><br/><a NAME="E3:74_1_1_2"/><i><font color="Green" title="E48">A7</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E4:74_1_1_2"/>
<font color="Maroon" title="c10">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E6:74_1_1_2"/><i><font color="Green" title="E49">A9</font></i>: 
( <font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E7:74_1_1_2"/>
<font color="Maroon" title="c11">A3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">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> <b> such that </b><br/><a NAME="E9:74_1_1_2"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c11">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E10:74_1_1_2"/><i><font color="Green" title="E51">A8</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A3</font> )
 
<b>by </b><i><a class="txt" href="#E3:74_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="#E6:74_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="#E9:74_1_1_2"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/>
<a NAME="E11:74_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c3">M</font> &amp; <span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">A2</font>,<font color="Maroon" title="c11">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font></span>)</span>,<font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>;<br/>
<a NAME="E12:74_1_1_2"/><b>then </b><i><font color="Green" title="E52">A15</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th01a</a>, <a class="txt" href="#E3:74_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="#E6:74_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="#E9:74_1_1_2"><i><font color="Green" title="E50">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c17">c1</font> = <font color="Maroon" title="c14">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="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:74_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="#E6:74_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="#E9:74_1_1_2"><i><font color="Green" title="E50">A11</font></i></a></i>;<br/>
<a NAME="E14:74_1_1_2"/><i><font color="Green" title="E53">A17</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">b1</font> &amp; <font color="Maroon" title="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">c1</font> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>;<br/>
<a NAME="E15:74_1_1_2"/><b>then </b>
<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="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c16">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>;<br/>
<a NAME="E16:74_1_1_2"/><b>then </b>
<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="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">c1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E17:74_1_1_2"/><b>then </b>
<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="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">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="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font>
 
<b>by </b><i><a class="txt" href="#E14:74_1_1_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>;<br/>
<a NAME="E18:74_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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"><font color="Maroon" title="c13">b</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="#E10:74_1_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="#E12:74_1_1_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>;<br/>
<b>hence </b><a NAME="E19:74_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:74_1_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="#E12:74_1_1_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:74_1_1"/><i><font color="Green" title="E48">A19</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">A1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #); <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">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> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font></span><br/>

<a NAME="E1:74_1_1_3"/>
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E3:74_1_1_3"/><i><font color="Green" title="E49">A20</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E4:74_1_1_3"/><i><font color="Green" title="E50">A21</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font>
 
<b>by </b><i><a class="txt" href="#E3:74_1_1_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c10">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="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:74_1_1_3"><i><font color="Green" title="E49">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">z</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> <b> such that </b><br/><a NAME="E7:74_1_1_3"/><i><font color="Green" title="E51">A22</font></i>: 
( <font color="Maroon" title="c12">z</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="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="ref" href="lpspace1.html#D16" target="_self" title="LPSPACE1:def.16">VSPDef4X</a></i>;<br/>
<a NAME="E8:74_1_1_3"/><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="#E7:74_1_1_3"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c14">z1</font> = <font color="Maroon" title="c12">z</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:74_1_1_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E7:74_1_1_3"><i><font color="Green" title="E51">A22</font></i></a></i>;<br/>
<font color="Maroon" title="c10">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c12">z</font> = 
<font color="Maroon" title="c13">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">z1</font>
<b>by </b><i><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>
<br/>.= 
<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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lpspace1.html#T34" target="_self" title="LPSPACE1:th.34">Th09a</a>, <a class="txt" href="#E7:74_1_1_3"><i><font color="Green" title="E51">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E11:74_1_1_3"/>
<font color="Maroon" title="c9">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> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font>
 <b>by </b><i><a class="txt" href="#E3:74_1_1_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E4:74_1_1_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="#E8:74_1_1_3"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:74_1_1"/><i><font color="Green" title="E49">A25</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">A1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #); <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">A1</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<a NAME="E1:74_1_1_4"/>
<font color="Maroon" title="c9">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E3:74_1_1_4"/><i><font color="Green" title="E50">A26</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E4:74_1_1_4"/><i><font color="Green" title="E51">A27</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A1</font>
 
<b>by </b><i><a class="txt" href="#E3:74_1_1_4"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c10">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="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:74_1_1_4"><i><font color="Green" title="E50">A26</font></i></a></i>;<br/>
<a NAME="E6:74_1_1_4"/><i><font color="Green" title="E52">A28</font></i>: 
<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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="#E3:74_1_1_4"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">A2</font> =  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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="c10">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>;<br/>
<a NAME="E7:74_1_1_4"/>
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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="c10">a</font></span>)</span>,<font color="Maroon" title="c3">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="#E6:74_1_1_4"><i><font color="Green" title="E52">A28</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">A2</font> =  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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="c10">a</font></span>)</span>,<font color="Maroon" title="c3">M</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c13">A2</font>
; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">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> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #)</span><br/>

<a NAME="E9:74_1_1_4"/><i><font color="Green" title="E53">A29</font></i>: 
<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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A2</font>
 
<b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="txt" href="#E3:74_1_1_4"><i><font color="Green" title="E50">A26</font></i></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">v</font>, <font color="Maroon" title="c15">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> <b> such that </b><br/><a NAME="E11:74_1_1_4"/><i><font color="Green" title="E54">A30</font></i>: 
( <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c11">a1</font></span>)</span> &amp; <font color="Maroon" title="c15">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="c14">v</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="ref" href="lpspace1.html#T27" target="_self" title="LPSPACE1:th.27">ThB11Z</a></i>;<br/>
<a NAME="E12:74_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="c10">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="c11">a1</font>
 
<b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>;<br/>
<a NAME="E13:74_1_1_4"/><b>then </b><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c10">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="c10">a</font></span>)</span> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a>, <a class="txt" href="#E11:74_1_1_4"><i><font color="Green" title="E54">A30</font></i></a></i>;<br/>
<a NAME="E14:74_1_1_4"/><i><font color="Green" title="E56">A32</font></i>: 
<font color="Maroon" title="c10">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="c10">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th01a</a>, <a class="txt" href="#E3:74_1_1_4"><i><font color="Green" title="E50">A26</font></i></a>, <a class="txt" href="#E6:74_1_1_4"><i><font color="Green" title="E52">A28</font></i></a></i>;<br/>
<a NAME="E15:74_1_1_4"/>
 ex <font color="Olive" title="b1">z</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>  st <br/>( <font color="Olive" title="b1">z</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="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Olive" title="b1">z</font>,<font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="lpspace1.html#D16" target="_self" title="LPSPACE1:def.16">VSPDef4X</a></i>;<br/>
<a NAME="E16:74_1_1_4"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c10">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="#E13:74_1_1_4"><i><font color="Green" title="E55">A31</font></i></a>, <a class="ref" href="lpspace1.html#T39" target="_self" title="LPSPACE1:th.39">EQC02</a>, <a class="txt" href="#E11:74_1_1_4"><i><font color="Green" title="E54">A30</font></i></a>, <a class="txt" href="#E14:74_1_1_4"><i><font color="Green" title="E56">A32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:74_1_1_4"/>
<font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">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> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="#E4:74_1_1_4"><i><font color="Green" title="E51">A27</font></i></a>, <a class="txt" href="#E9:74_1_1_4"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) holds <br/> ( <font color="Maroon" title="c9">x</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="c9">x</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="c9">x</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="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</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="c9">x</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="c10">y</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="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</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="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</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/><b>let </b><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 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">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="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">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="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font> )</span><br/><a NAME="E1:74_1_1_5"/>
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:74_1_1_5"/><i><font color="Green" title="E50">A35</font></i>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/><a NAME="E4:74_1_1_5"/>
<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E6:74_1_1_5"/><i><font color="Green" title="E51">A37</font></i>: 
( <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c3">M</font> )
 ;<br/><a NAME="E7:74_1_1_5"/><i><font color="Green" title="E52">A36</font></i>: 
( <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> )
 <b>by </b><i><a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="#E6:74_1_1_5"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/><a NAME="E8:74_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</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="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</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>
 <b>by </b><i><a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>;<br/><a NAME="E9:74_1_1_5"/><b>then </b><i><font color="Green" title="E53">A40</font></i>: 
<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> <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>
 <b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th01a</a>, <a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="#E6:74_1_1_5"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/><a NAME="E10:74_1_1_5"/><i><font color="Green" title="E54">A41</font></i>: 
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c14">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="lpspace1.html#K10" title="LPSPACE1:func.10">RLSp_L1Funct</a> <font color="Maroon" title="c3">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="#E6:74_1_1_5"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/><a NAME="E12:74_1_1_5"/><i><font color="Green" title="E55">A42</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">b1</font> )
 <b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>;<br/><a NAME="E13:74_1_1_5"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">b1</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>;<br/><a NAME="E14:74_1_1_5"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">b1</font></span>)</span>
 <b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>;<br/><a NAME="E15:74_1_1_5"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">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="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">b1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><a NAME="E16:74_1_1_5"/><b>then </b><i><font color="Green" title="E56">A45</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</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> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">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="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:74_1_1_5"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>;<br/><i><font color="Green" title="E57">A46</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font>
<b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">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="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">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="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:74_1_1_5"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">ThB10</a></i>
;<br/><i><font color="Green" title="E58">A47</font></i>: 1 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> = 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font>
<b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>
<br/>.= 
<font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/><i><font color="Green" title="E59">A48</font></i>: <font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span> = 
<font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:74_1_1_5"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">a1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">ThB11</a></i>
;<br/><a NAME="E20:74_1_1_5"/><i><font color="Green" title="E60">A49</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="txt" href="#E10:74_1_1_5"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a>, <a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a></i>;<br/><a NAME="E21:74_1_1_5"/>
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><a NAME="E22:74_1_1_5"/><b>then </b><i><font color="Green" title="E61">A51</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A2</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a>, <a class="txt" href="#E6:74_1_1_5"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/><a NAME="E23:74_1_1_5"/>
<font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">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="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c14">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E16:74_1_1_5"><i><font color="Green" title="E56">A45</font></i></a>, <a class="txt" href="#E9:74_1_1_5"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><b>hence </b><a NAME="E24:74_1_1_5"/>
<font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">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="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:74_1_1_5"><i><font color="Green" title="E60">A49</font></i></a>, <a class="txt" href="#E22:74_1_1_5"><i><font color="Green" title="E61">A51</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">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="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font> )</span><br/><a NAME="E25:74_1_1_5"/>
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><a NAME="E26:74_1_1_5"/><b>then </b><i><font color="Green" title="E62">A53</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a>, <a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a></i>;<br/><a NAME="E27:74_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">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="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E17:74_1_1_5"><i><font color="Green" title="E57">A46</font></i></a>, <a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><b>hence </b><a NAME="E28:74_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">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="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:74_1_1_5"><i><font color="Green" title="E60">A49</font></i></a>, <a class="txt" href="#E26:74_1_1_5"><i><font color="Green" title="E62">A53</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span> &amp; 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font> )</span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> = 
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="txt" href="#E19:74_1_1_5"><i><font color="Green" title="E59">A48</font></i></a>, <a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E26:74_1_1_5"><i><font color="Green" title="E62">A53</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font></span><br/><b>thus </b><a NAME="E30:74_1_1_5"/>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font>
 <b>by </b><i><a class="txt" href="#E3:74_1_1_5"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="#E18:74_1_1_5"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E7:74_1_1_5"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">A</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font></span>)</span> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a>  <b>by </b><i><a class="txt" href="#E1:74_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="#E2:74_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="#E3:74_1_1"><i><font color="Green" title="E48">A19</font></i></a>, <a class="txt" href="#E4:74_1_1"><i><font color="Green" title="E49">A25</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c9">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c9">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> )</span><br/>

<b>thus </b><a NAME="E7:74_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c9">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c9">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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