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

<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">aC</font> =  <a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">zC</font> =  <a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">lC</font> =  <a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">A</font> =  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #);<br/>
<a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">A1</font>, <font color="Maroon" title="c11">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A1</font></span><br/>

<a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c10">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
( <font color="Maroon" title="c10">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c4">M</font>) &amp; <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/>
<a NAME="E4:70_1_1_1"/>
<font color="Maroon" title="c11">A2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E6:70_1_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
( <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c4">M</font>) &amp; <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/>
<a NAME="E7:70_1_1_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E8:70_1_1_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E9:70_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:70_1_1_1"/>
<font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E7:70_1_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>;<br/>
<a NAME="E1:70_1_1_2"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lpspacc1.html#E29"><span class="lab"><font color="Green" title="E18">Lm3</font></span></a></span>;<br/>
<a NAME="E2:70_1_1_2"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_2"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">A1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font></span><br/>

<a NAME="E3:70_1_1_2"/>
<font color="Maroon" title="c11">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E5:70_1_1_2"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E6:70_1_1_2"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c12">a</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_2"><span class="lab"><font color="Green" title="E45">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">z1</font> = <font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_2"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E1:70_1_1_2"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A11</font></span>: <font color="Maroon" title="c12">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></span>)</span> = 
<font color="Maroon" title="c14">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<font color="Maroon" title="c14">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="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/>
<a NAME="E10:70_1_1_2"/>
<font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E6:70_1_1_2"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:70_1_1_2"/>
<font color="Maroon" title="c11">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E2:70_1_1_2"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E9:70_1_1_2"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a><br/>  for <font color="Olive" title="b3">A1</font>, <font color="Olive" title="b4">A2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b1">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b4">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) holds <br/> ( <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_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="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_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">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_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="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_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="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">A1</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">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="c12">A1</font>, <font color="Maroon" title="c13">A2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">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">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/><a NAME="E2:70_1_1_3"/>
<font color="Maroon" title="c12">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E4:70_1_1_3"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E5:70_1_1_3"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/><a NAME="E6:70_1_1_3"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E7:70_1_1_3"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c12">A1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>;<br/><a NAME="E8:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E9:70_1_1_3"/>
<font color="Maroon" title="c13">A2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E11:70_1_1_3"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<font color="Maroon" title="c13">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c17">b</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E12:70_1_1_3"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
<font color="Maroon" title="c17">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">a1</font> = <font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c19">b1</font> = <font color="Maroon" title="c17">b</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a></span>;<br/><a NAME="E14:70_1_1_3"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E15:70_1_1_3"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
<font color="Maroon" title="c17">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_1_3"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E16:70_1_1_3"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c13">A2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c17">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_1_3"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>;<br/><a NAME="E17:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c17">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E18:70_1_1_3"/>
<font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/><a NAME="E19:70_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E20:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">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">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c19">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D2" onmouseover="rs('clvect_1/D2')" onmouseout="rh()">CLVECT_1:def 2</a></span>;<br/><a NAME="E21:70_1_1_3"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c19">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E22:70_1_1_3"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">A1</font>,<font color="Maroon" title="c13">A2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_3"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E15:70_1_1_3"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/><a NAME="E23:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E12:70_1_1_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font></span>)</span>)

<br/>.= 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1_1_3"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_1_3"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E20:70_1_1_3"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E21:70_1_1_3"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>
;<br/><span class="kw">hence </span><a NAME="E25:70_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">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">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_3"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E17:70_1_1_3"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A1</font> )</span><br/><a NAME="E26:70_1_1_3"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>;<br/><a NAME="E27:70_1_1_3"/>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c12">A1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>;<br/><a NAME="E28:70_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">y1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/><span class="lab"><font color="Green" title="E57">A27</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">y1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">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="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D3" onmouseover="rs('clvect_1/D3')" onmouseout="rh()">CLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:70_1_1_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E14:70_1_1_3"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">y1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> = 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">y1</font></span>)</span>,<font color="Maroon" title="c12">A1</font>)

<br/>.= 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:70_1_1_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>
;<br/><span class="kw">hence </span><a NAME="E31:70_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_3"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E28:70_1_1_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span> &amp; <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A1</font> )</span><br/><span class="lab"><font color="Green" title="E58">A28</font></span>: <font color="Maroon" title="c14">x1</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span> = 
<font color="Maroon" title="c14">x1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:70_1_1_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D4" onmouseover="rs('clvect_1/D4')" onmouseout="rh()">CLVECT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y1</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> = 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y1</font></span>)</span>,<font color="Maroon" title="c12">A1</font>)

<br/>.= 
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">x1</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:70_1_1_3"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c14">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">y1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:70_1_1_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
<font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E34:70_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A1</font></span><br/><span class="lab"><font color="Green" title="E59">A29</font></span>: <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c16">a</font> = 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c18">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<font color="Maroon" title="c16">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" onmouseover="rs('clvect_1/D5')" onmouseout="rh()">CLVECT_1:def 5</a></span>
;<br/><span class="kw">thus </span><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">A1</font> = 
<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>,<font color="Maroon" title="c12">A1</font>)

<br/>.= 
<font color="Maroon" title="c12">A1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_3"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E35:70_1_1_3"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E5:70_1_1_3"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a>, <a class="ref" href="lpspacc1.html#D18" target="_self" onmouseover="rs('lpspacc1/D18')" onmouseout="rh()">Def18</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:70_1_1"/><span class="lab"><font color="Green" title="E43">A30</font></span>: 
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><span class="kw">proof </span></a><div class="add">

<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="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">A1</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<a NAME="E1:70_1_1_4"/>
<font color="Maroon" title="c10">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">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="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_4"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
<font color="Maroon" title="c10">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_4"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">A2</font> =  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>);<br/>
<a NAME="E5:70_1_1_4"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E6:70_1_1_4"/><span class="kw">then </span>
 <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">A2</font> =  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span>,<font color="Maroon" title="c4">M</font>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) ;<br/>
<a NAME="E8:70_1_1_4"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
( <font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">A1</font> &amp; <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_4"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="lpspacc1.html#T18" target="_self" onmouseover="rs('lpspacc1/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c11">a</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E45">A32</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">A2</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">v</font>, <font color="Maroon" title="c16">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E11:70_1_1_4"/>
<font color="Maroon" title="c15">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E12:70_1_1_4"/>
<font color="Maroon" title="c16">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E13:70_1_1_4"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:70_1_1_4"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>
 <span class="kw">and </span><br/><a NAME="E15:70_1_1_4"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T21" target="_self" onmouseover="rs('lpspacc1/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E16:70_1_1_4"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lpspacc1.html#E29"><span class="lab"><font color="Green" title="E18">Lm3</font></span></a></span>;<br/>
<a NAME="E17:70_1_1_4"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_4"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E5:70_1_1_4"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E18:70_1_1_4"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T20" target="_self" onmouseover="rs('lpspacc1/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E19:70_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lpspacc1.html#R1" title="LPSPACC1:pred.1">a.e.cpfunc=</a> <font color="Maroon" title="c16">g</font>,<font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_1_4"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E15:70_1_1_4"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E20:70_1_1_4"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_1_4"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E17:70_1_1_4"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E16:70_1_1_4"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="lpspacc1.html#T32" target="_self" onmouseover="rs('lpspacc1/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:70_1_1_4"/>
<font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_4"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:70_1_1"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">A1</font>, <font color="Maroon" title="c11">A2</font>, <font color="Maroon" title="c12">A3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c11">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font></span>)</span></span><br/>

<a NAME="E1:70_1_1_5"/>
<font color="Maroon" title="c10">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_5"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
<font color="Maroon" title="c10">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_5"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/>
<a NAME="E5:70_1_1_5"/>
<font color="Maroon" title="c11">A2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E7:70_1_1_5"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
<font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E8:70_1_1_5"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/>
<a NAME="E9:70_1_1_5"/>
<font color="Maroon" title="c12">A3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">c</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E11:70_1_1_5"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c12">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c4">M</font>)
 <span class="kw">and </span><br/><a NAME="E12:70_1_1_5"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lpspacc1.html#K13" title="LPSPACC1:func.13">L1_CFunctions</a> <font color="Maroon" title="c4">M</font>
 ;<br/>
<a NAME="E13:70_1_1_5"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_1_5"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E12:70_1_1_5"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E14:70_1_1_5"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1_5"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E8:70_1_1_5"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E15:70_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">A2</font>,<font color="Maroon" title="c12">A3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_1_5"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<a NAME="E16:70_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_5"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E12:70_1_1_5"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">a1</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c17">b1</font> = <font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c18">c1</font> = <font color="Maroon" title="c15">c</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lpspacc1.html#K14" title="LPSPACC1:func.14">CLSp_L1Funct</a> <font color="Maroon" title="c4">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_5"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E8:70_1_1_5"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E12:70_1_1_5"><span class="lab"><font color="Green" title="E49">A45</font></span></a></span>;<br/>
<a NAME="E18:70_1_1_5"/>
<font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E19:70_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E20:70_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A49</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>;<br/>
<a NAME="E21:70_1_1_5"/>
<font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E22:70_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A50</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">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="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:70_1_1_5"><span class="lab"><font color="Green" title="E53">A49</font></span></a>, <a class="ref" href="lpspacc1.html#T19" target="_self" onmouseover="rs('lpspacc1/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E23:70_1_1_5"/><span class="lab"><font color="Green" title="E55">A51</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_5"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E4:70_1_1_5"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E24:70_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">A1</font>,<font color="Maroon" title="c11">A2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c14">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_1_5"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<a NAME="E25:70_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c14">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_5"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E8:70_1_1_5"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="ref" href="lpspacc1.html#T17" target="_self" onmouseover="rs('lpspacc1/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="lpspacc1.html#T31" target="_self" onmouseover="rs('lpspacc1/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E26:70_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K17" title="LPSPACC1:func.17">a.e-Ceq-class</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70_1_1_5"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E22:70_1_1_5"><span class="lab"><font color="Green" title="E54">A50</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:70_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">A1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">A2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c11">A2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">A3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1_1_5"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E16:70_1_1_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="lpspacc1.html#D16" target="_self" onmouseover="rs('lpspacc1/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> =  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <span class="p1">(<span class="default"><a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">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#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="clvect_1.html#V1" title="CLVECT_1:attr.1">strict</a>   <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">vector-distributive</a>   <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">scalar-distributive</a>   <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">scalar-associative</a>   <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">scalar-unital</a>   <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E3:70_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="clvect_1.html#D2" onmouseover="rs('clvect_1/D2')" onmouseout="rh()">CLVECT_1:def 2</a>, <a class="ref" href="clvect_1.html#D3" onmouseover="rs('clvect_1/D3')" onmouseout="rh()">CLVECT_1:def 3</a>, <a class="ref" href="clvect_1.html#D4" onmouseover="rs('clvect_1/D4')" onmouseout="rh()">CLVECT_1:def 4</a>, <a class="ref" href="clvect_1.html#D5" onmouseover="rs('clvect_1/D5')" onmouseout="rh()">CLVECT_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">A</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c10">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font> &amp;  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c10">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:70_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K18" title="LPSPACC1:func.18">CCosetSet</a> <font color="Maroon" title="c4">M</font> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c10">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspacc1.html#K19" title="LPSPACC1:func.19">addCCoset</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspacc1.html#K20" title="LPSPACC1:func.20">zeroCCoset</a> <font color="Maroon" title="c4">M</font> &amp;  the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c10">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspacc1.html#K21" title="LPSPACC1:func.21">lmultCCoset</a> <font color="Maroon" title="c4">M</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
