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

<b>let </b><font color="Maroon">GF</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">V1</font> is <a href="vectsp_4.html#V1" target="_self">lineary-closed</a>
 ;<br/>

<b>set </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">V1</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="vectsp_4.html#E2:37"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font>;<br/>
<a NAME="E4:37"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</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">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</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="E5:37"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</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">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E4:37"><i><font color="Green">E29</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:37"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T7" target="_self">Th7</a>, , <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:37_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T7" target="_self">Th7</a>, , <a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E9:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, , <a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T5" target="_self">Th5</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">D</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font>;<br/>
<a NAME="E9:37"/>
( <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E11:37"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><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="E1:37_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25">VECTSP_1:def 25</a></i>
;<br/>
<b>hence </b><a NAME="E7:37_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T5" target="_self">Th5</a>, , , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font> as   <a href="binop_1.html#NM1">UnOp</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">M</font> = the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</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">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">d</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E14:37"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</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">GF</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</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="E15:37"/>
<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">GF</font>,<font color="Maroon">D</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">GF</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E16:37"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>,<font color="Maroon">D</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 <font color="Maroon">GF</font>,<font color="Maroon">D</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/>
<a NAME="E17:37"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><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="E1:37_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_3"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37_3"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E7:37_3"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E10:37_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:37_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T5" target="_self">Th5</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</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">GF</font>,<font color="Maroon">D</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 <font color="Maroon">GF</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #);<br/>
<a NAME="E19:37"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #)<br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green">E108</font></i>: 
(  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_4"/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</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">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #);<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E2:37_4_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
<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="rlvect_1.html#K4">+</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</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">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #);<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon">a</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>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #)</span>)</span> = 
<font color="Maroon">x</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">V</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</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">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) ;<br/>
<a NAME="E8:37_4"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">V</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25">VECTSP_1:def 25</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:37"/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) is <a href="vectsp_1.html#V12">VectSp-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>; <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="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #), <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #);<br/>



<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">v</font>, <font color="Maroon">y</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #);<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:37_5_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:37_5_1"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green">E33</font></i></a></i>;<br/><b>thus </b><a NAME="E3:37_5_1"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:37_5"/><b>then </b><i><font color="Green">E115</font></i>: 
( <font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font> )
 
;<br/>
<a NAME="E4:37_5"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</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">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">a</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">a</font> <a href="vectsp_1.html#K6">*</a> <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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</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">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</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">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# <font color="Maroon">D</font>,<font color="Maroon">A</font>,<font color="Maroon">d</font>,<font color="Maroon">M</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E24:37"/>
<font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>


</div>
