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

<b>set </b><font color="Maroon">cR</font> =  <a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">ccR</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">ccs</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">lm</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:44_1_1"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:44_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:44_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:44_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:44_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E6:44_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:44_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:44_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:44_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:44_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E8:44_1_1_2"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E9:44_1_1_2"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">lm</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>set </b><font color="Maroon">Vos</font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #);<br/>
<b>set </b><font color="Maroon">cV</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #);<br/>
<b>set </b><font color="Maroon">aV</font> = the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #);<br/>
<b>set </b><font color="Maroon">ac</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">ZV</font> = the <a href="struct_0.html#U2">Zero</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #);<br/>
<b>set </b><font color="Maroon">Zc</font> = the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>;<br/>
<a NAME="E7:44_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) is <a href="vectsp_1.html#V12">VectSp-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a><br/><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #), <font color="Maroon">w</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #);<br/>



<a NAME="E1:44_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font>, <font color="Maroon">ww</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E4:44_1_1_3"/><i><font color="Green">E57</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:44_1_1_3"/><i><font color="Green">E58</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:44_1_1_3"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:44_1_1_3"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E8:44_1_1_3"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E9:44_1_1_3"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E10:44_1_1_3"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:44_1_1_3"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:44_1_1_3"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E13:44_1_1_3"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E14:44_1_1_3"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">vv</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">ww</font></span>)</span>
<b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <font color="Maroon">ww</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xx</font>,<font color="Maroon">ww</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">vv</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">yy</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">yy</font>,<font color="Maroon">vv</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">vv</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="weddwitt.html#E4:44_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <font color="Maroon">yy</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">xx</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xx</font>,<span class="p2">(<span class="default"><font color="Maroon">lm</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">v</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>

<a NAME="E18:44_1_1_3"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="rlvect_1.html#R1">in</a>  <a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:44_1_1_3"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E20:44_1_1_3"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>
 
;<br/>
<a NAME="E21:44_1_1_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">vv</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">lm</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">vv</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">vv</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:44_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) is <a href="rlvect_1.html#V4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #), <font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #), <font color="Maroon">w</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon">uu</font> = <font color="Maroon">u</font>, <font color="Maroon">vv</font> = <font color="Maroon">v</font>, <font color="Maroon">ww</font> = <font color="Maroon">w</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">uu</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">vv</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">ww</font>

<br/>.= 
<font color="Maroon">uu</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">vv</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">ww</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E9:44_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) is <a href="rlvect_1.html#V5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #)</span>)</span> = 
<font color="Maroon">vv</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>

<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:44_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">ww</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:44_1_1_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">vv</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) ;<br/>
<a NAME="E5:44_1_1_6"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #)
 
<b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:44_1_1_6"/>
 ex <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) st <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:44_1_1"/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #) is <a href="rlvect_1.html#V3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #), <font color="Maroon">w</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,<font color="Maroon">lm</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font>, <font color="Maroon">ww</font> = <font color="Maroon">w</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>

<br/>.= 
<font color="Maroon">ww</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">vv</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:44_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#NM6">VectSp</a> of  <a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font> st <br/>(  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #) &amp; the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, , , </i>;<br/>


</div>
