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

<b>let </b><font color="Maroon" title="c1">GF</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">VectSp-like</a>  <a href="vectsp_1.html#L4" title="VECTSP_1:struct.4">VectSpStr</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c3">V1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">V1</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c3">V1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="vectsp_4.html#E1:37"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">A</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c3">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:37"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E4:37"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E7:37"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_1"/><i><font color="Green" title="E31">A6</font></i>: 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:37_1"/><i><font color="Green" title="E32">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:37_1"/><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="vectsp_4.html#E6:37_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c12">y2</font> = <font color="Maroon" title="c10">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E9:37_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">y2</font>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E3:37_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="vectsp_4.html#E4:37_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="vectsp_4.html#E6:37_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:37_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:37"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E7:37_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="vectsp_4.html#D1" target="_self" title="VECTSP_4:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">D</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c5">D</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">C</font> = <span class="p1">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font>;<br/>
<a NAME="E9:37"/>
( <font color="Maroon" title="c3">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E11:37"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">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" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2"/><i><font color="Green" title="E32">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E10:37"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_2"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/>
<font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E3:37_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="vectsp_4.html#E4:37_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25" title="VECTSP_1:def.25">VECTSP_1:def 25</a></i>
;<br/>
<b>hence </b><a NAME="E7:37_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:37"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E10:37"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="vectsp_4.html#T5" target="_self" title="VECTSP_4:th.5">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">C</font> = <span class="p1">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">D</font> as   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c5">D</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E10:37"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">M</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">d</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E2:37"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="vectsp_4.html#T4" target="_self" title="VECTSP_4:th.4">Th4</a></i>;<br/>
<a NAME="E14:37"/><i><font color="Green" title="E31">A12</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:37"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E16:37"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E14:37"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E17:37"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">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" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_3"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_3"/><i><font color="Green" title="E34">A15</font></i>: 
<span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y1</font>, <font color="Maroon" title="c15">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:37_3"/><i><font color="Green" title="E35">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y1</font>,<font color="Maroon" title="c15">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E16:37"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:37_3"/><i><font color="Green" title="E36">A17</font></i>: 
( <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font> &amp; <font color="Maroon" title="c15">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E16:37"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="vectsp_4.html#E6:37_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c14">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E16:37"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="vectsp_4.html#E6:37_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y2</font> = <font color="Maroon" title="c15">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E7:37_3"><i><font color="Green" title="E36">A17</font></i></a></i>;<br/>
<a NAME="E10:37_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c17">y2</font>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E3:37_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="vectsp_4.html#E4:37_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="vectsp_4.html#E6:37_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:37_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:37"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E7:37_3"><i><font color="Green" title="E36">A17</font></i></a>, <a class="ref" href="vectsp_4.html#D1" target="_self" title="VECTSP_4:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">M</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GF</font>,<font color="Maroon" title="c5">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c5">D</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E16:37"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">W</font> =  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/>
<a NAME="E19:37"/><i><font color="Green" title="E33">A18</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #)<br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/><b>let </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>;<br/>



<b>assume </b><a NAME="E1:37_4"/><i><font color="Green" title="E34">A19</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">b</font> = 
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<font color="Maroon" title="c16">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E6:37"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="txt" href="vectsp_4.html#E1:37_4"><i><font color="Green" title="E34">A19</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:37"/><i><font color="Green" title="E34">A20</font></i>: 
(  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_5"/>
 <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c15">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">b</font> = 
<font color="Maroon" title="c17">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></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" title="RLVECT_1:def.6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font>, <font color="Maroon" title="c16">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/><b>reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c15">b</font>, <font color="Maroon" title="c19">z</font> = <font color="Maroon" title="c16">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E2:37_5_2"/><i><font color="Green" title="E35">A21</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">y</font> &amp; <font color="Maroon" title="c15">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">z</font> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c16">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c16">c</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="vectsp_4.html#E2:37_5_2"><i><font color="Green" title="E35">A21</font></i></a></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" title="RLVECT_1:def.7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/><b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #)</span>)</span> = 
<font color="Maroon" title="c15">x</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> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a'</font> = <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c9">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c17">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) ;<br/>
<a NAME="E8:37_5"/><i><font color="Green" title="E35">A22</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E10:37"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c18">b</font>
;<br/>

<b>thus </b><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c18">b</font> = 
<font color="Maroon" title="c15">x</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="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="vectsp_4.html#E8:37_5"><i><font color="Green" title="E35">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25" title="VECTSP_1:def.25">VECTSP_1:def 25</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:37"/>
 <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) is <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">VectSp-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a><br/><b>let </b><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c17">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><i><font color="Green" title="E35">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c21">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #);<br/><b>let </b><font color="Maroon" title="c22">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>;<br/><b>assume </b><a NAME="E1:37_6_1"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font>
 ;<br/><a NAME="E2:37_6_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">a</font>,<font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">M</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E16:37"><i><font color="Green" title="E32">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_6_1"/>
<font color="Maroon" title="c20">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E1:37_6_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:37_6"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
( <font color="Maroon" title="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font> &amp; <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c19">y</font> )
 
;<br/>
<a NAME="E4:37_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c17">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</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="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c19">y</font></span>)</span> &amp; <font color="Maroon" title="c16">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">y</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c17">w</font></span>)</span> = 
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c19">y</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E2:37_6"><i><font color="Green" title="E35">A23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</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="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c19">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c17">w</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_6"><i><font color="Green" title="E36">A25</font></i></a></i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E2:37_6"><i><font color="Green" title="E35">A23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</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="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E19:37"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_6"><i><font color="Green" title="E36">A25</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E2:37_6"><i><font color="Green" title="E35">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_4.html#E2:37_6"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="vectsp_4.html#E3:37_6"><i><font color="Green" title="E36">A25</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c16">v</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">x</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E2:37_6"><i><font color="Green" title="E35">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">W</font> =  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">M</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">VectSp-like</a>  <a href="vectsp_1.html#L4" title="VECTSP_1:struct.4">VectSpStr</a> of <font color="Maroon" title="c1">GF</font> <b>by </b><i><a class="txt" href="vectsp_4.html#E20:37"><i><font color="Green" title="E34">A20</font></i></a></i>;<br/>
<a NAME="E23:37"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c14">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">W</font> = <font color="Maroon" title="c14">W</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#D2" target="_self" title="VECTSP_4:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">W</font>
;<br/>

<b>thus </b><a NAME="E25:37"/>
<font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">W</font>
 ;<br/>


</div>
