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

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 <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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:74_1_1_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
( <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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E6:74_1_1_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
( <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"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E8:74_1_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E9:74_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:74_1_1_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:74_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E7:74_1_1_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:74_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
 <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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E2:74_1_1_2"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c9">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> 
 <span class="kw">and </span><br/><a NAME="E3:74_1_1_2"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">and </span><br/><a NAME="E4:74_1_1_2"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
 <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="c9">z</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D16" target="_self" title="LPSPACE1:def.16">Def16</a></span>;<br/>
<a NAME="E5:74_1_1_2"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c9">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> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_2"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E4:74_1_1_2"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">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> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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="c10">A1</font></span><br/>

<a NAME="E6:74_1_1_2"/>
<font color="Maroon" title="c10">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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:74_1_1_2"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">and </span><br/><a NAME="E9:74_1_1_2"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:74_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c14">z1</font> = <font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:74_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E3:74_1_1_2"><span class="lab"><font color="Green" title="E49">A8</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A13</font></span>: <font color="Maroon" title="c11">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c9">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_1_1_2"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="lpspace1.html#T34" target="_self" title="LPSPACE1:th.34">Th34</a></span>
<br/>.= 
<font color="Maroon" title="c11">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<a NAME="E13:74_1_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_2"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E9:74_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:74_1_1_2"/>
<font color="Maroon" title="c10">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="c10">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_2"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E5:74_1_1_2"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E12:74_1_1_2"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E48">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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/><span class="kw">let </span><font color="Maroon" title="c11">A1</font>, <font color="Maroon" title="c12">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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/><span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c10">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E2:74_1_1_3"/>
<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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:74_1_1_3"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<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="c15">a</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E5:74_1_1_3"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c15">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="E6:74_1_1_3"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_3"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E5:74_1_1_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><a NAME="E7:74_1_1_3"/><span class="kw">then </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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</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="c15">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>;<br/><a NAME="E8:74_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_1_1_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th24</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><a NAME="E9:74_1_1_3"/>
<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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E11:74_1_1_3"/><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<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="c16">b</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E12:74_1_1_3"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<font color="Maroon" title="c16">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/><span class="kw">reconsider </span><font color="Maroon" title="c17">a1</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c18">b1</font> = <font color="Maroon" title="c16">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_1_1_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E12:74_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a></span>;<br/><a NAME="E14:74_1_1_3"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>;<br/><a NAME="E15:74_1_1_3"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:74_1_1_3"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="txt" href="#E12:74_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><a NAME="E16:74_1_1_3"/><span class="kw">then </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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</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="c16">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>;<br/><a NAME="E17:74_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A23</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:74_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th24</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><a NAME="E18:74_1_1_3"/>
<font color="Maroon" title="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/><a NAME="E19:74_1_1_3"/><span class="kw">then </span>
<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="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>;<br/><a NAME="E20:74_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A24</font></span>: 
<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="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">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="c17">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="c18">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>;<br/><a NAME="E21:74_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c18">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>;<br/><a NAME="E22:74_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A25</font></span>: 
<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="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">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="c15">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="c16">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74_1_1_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E20:74_1_1_3"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/><a NAME="E23:74_1_1_3"/>
<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="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="#E15:74_1_1_3"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>;<br/><a NAME="E24:74_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_1_1_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E12:74_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th23</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><a NAME="E25:74_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">x1</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="c15">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="c16">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:74_1_1_3"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E26:74_1_1_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E17:74_1_1_3"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="E27:74_1_1_3"/><span class="lab"><font color="Green" title="E60">A26</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>;<br/><a NAME="E28:74_1_1_3"/>
<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="c14">y1</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="c15">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>;<br/><a NAME="E29:74_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A27</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_1_1_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th24</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c15">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">y1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>
<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="c17">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="c17">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></span>
<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="c15">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="c15">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:74_1_1_3"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E14:74_1_1_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>
;<br/><a NAME="E31:74_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">y1</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="c15">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="c15">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E32:74_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E29:74_1_1_3"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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/><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="c15">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="c17">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:74_1_1_3"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E14:74_1_1_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">y1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">y1</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="c13">x1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">y1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:74_1_1_3"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>
;<br/><span class="kw">hence </span><a NAME="E35:74_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font></span><br/>1 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c15">a</font> = 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c17">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>
<br/>.= 
<font color="Maroon" title="c15">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
;<br/><span class="kw">hence </span><a NAME="E37:74_1_1_3"/>
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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_3"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E6:74_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:74_1_1"/><span class="lab"><font color="Green" title="E49">A28</font></span>: 
 <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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:74_1_1_4"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:74_1_1_4"/><span class="lab"><font color="Green" title="E51">A30</font></span>: 
<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/>
<span class="kw">set </span><font color="Maroon" title="c11">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="E5:74_1_1_4"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_4"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th24</a></span>;<br/>
<a NAME="E6:74_1_1_4"/><span class="kw">then </span>
 <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>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> 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/>
<a NAME="E8:74_1_1_4"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
( <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> &amp; <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="c12">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_4"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="txt" href="#E4:74_1_1_4"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th24</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_4"><span class="lab"><font color="Green" title="E51">A30</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">A2</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="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/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E11:74_1_1_4"/>
<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>
 <span class="kw">and </span><br/><a NAME="E12:74_1_1_4"/>
<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>
 <span class="kw">and </span><br/><a NAME="E13:74_1_1_4"/><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">a1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:74_1_1_4"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E15:74_1_1_4"/><span class="lab"><font color="Green" title="E56">A35</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T27" target="_self" title="LPSPACE1:th.27">Th27</a></span>;<br/>
<a NAME="E16:74_1_1_4"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
 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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D16" target="_self" title="LPSPACE1:def.16">Def16</a></span>;<br/>
<a NAME="E17:74_1_1_4"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_4"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E5:74_1_1_4"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th23</a></span>;<br/>
<a NAME="E18:74_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="c13">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T26" target="_self" title="LPSPACE1:th.26">Th26</a></span>;<br/>
<a NAME="E19:74_1_1_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:74_1_1_4"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E15:74_1_1_4"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/>
<a NAME="E20:74_1_1_4"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74_1_1_4"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E17:74_1_1_4"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E16:74_1_1_4"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="lpspace1.html#T39" target="_self" title="LPSPACE1:th.39">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:74_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="c12">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_4"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:74_1_1"/>
 <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_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="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_5"/>
<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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:74_1_1_5"/><span class="lab"><font color="Green" title="E50">A38</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:74_1_1_5"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
<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="E5:74_1_1_5"/>
<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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E7:74_1_1_5"/><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:74_1_1_5"/><span class="lab"><font color="Green" title="E53">A41</font></span>: 
<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="E9:74_1_1_5"/>
<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/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E11:74_1_1_5"/><span class="lab"><font color="Green" title="E54">A42</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:74_1_1_5"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<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="E13:74_1_1_5"/><span class="lab"><font color="Green" title="E56">A44</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:74_1_1_5"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E12:74_1_1_5"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E14:74_1_1_5"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:74_1_1_5"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="txt" href="#E8:74_1_1_5"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E15:74_1_1_5"/><span class="kw">then </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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:74_1_1_5"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>;<br/>
<a NAME="E16:74_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A46</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:74_1_1_5"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E12:74_1_1_5"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th23</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">a1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c16">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_5"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E8:74_1_1_5"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E12:74_1_1_5"><span class="lab"><font color="Green" title="E55">A43</font></span></a></span>;<br/>
<a NAME="E18:74_1_1_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/>
<a NAME="E19:74_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/>
<a NAME="E20:74_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E21:74_1_1_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/>
<a NAME="E22:74_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A48</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:74_1_1_5"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="lpspace1.html#T25" target="_self" title="LPSPACE1:th.25">Th25</a></span>;<br/>
<a NAME="E23:74_1_1_5"/><span class="lab"><font color="Green" title="E61">A49</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:74_1_1_5"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="txt" href="#E4:74_1_1_5"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E24:74_1_1_5"/><span class="kw">then </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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74_1_1_5"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>;<br/>
<a NAME="E25:74_1_1_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_1_1_5"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E8:74_1_1_5"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th23</a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">Th38</a></span>;<br/>
<a NAME="E26:74_1_1_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:74_1_1_5"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E22:74_1_1_5"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:74_1_1_5"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E23:74_1_1_5"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="txt" href="#E16:74_1_1_5"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:74_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E4:74_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E3:74_1_1"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
