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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c6">m2</font>, <font color="Maroon" title="c7">m</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">M</font>;<br/><b>let </b><font color="Maroon" title="c8">n1</font>, <font color="Maroon" title="c9">n2</font>, <font color="Maroon" title="c10">n</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c4">N</font>;<br/>



















<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font>
 ;<br/>

<b>thus </b><a NAME="E2:21"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:21_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c11">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:21"/>
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">n2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m1'</font> = <font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c13">m2'</font> = <font color="Maroon" title="c6">m2</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:21_2"/>
( <font color="Maroon" title="c12">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c13">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_2"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
<font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>;<br/>


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

<b>thus </b><a NAME="E4:21"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c10">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m'</font> = <font color="Maroon" title="c7">m</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:21_3"/>
<font color="Maroon" title="c12">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_3"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_3"/>
<font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>


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

<b>thus </b><a NAME="E5:21"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_4"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m'</font> = <font color="Maroon" title="c7">m</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:21_4"/>
<font color="Maroon" title="c12">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_4"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_4"/>
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T23" title="VECTSP_4:th.23">VECTSP_4:23</a></i>;<br/>


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

<b>thus </b><a NAME="E6:21"/>
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">n2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m1'</font> = <font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c13">m2'</font> = <font color="Maroon" title="c6">m2</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:21_5"/>
( <font color="Maroon" title="c12">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c13">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_5"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_5"/>
<font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</a></i>;<br/>


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

<b>thus </b><a NAME="E7:21"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:21_6"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<b>hence </b><a NAME="E3:21_6"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">N</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>


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

<b>thus </b><a NAME="E8:21"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">M</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:21_7"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c11">M'</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T27" title="VECTSP_4:th.27">VECTSP_4:27</a></i>;<br/>
<b>hence </b><a NAME="E3:21_7"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">M</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 ;<br/>


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

<b>thus </b><a NAME="E9:21"/>
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_8"/><i><font color="Green" title="E12">A6</font></i>: 
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:21_8"/>
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_8"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_8"/>
<font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</a></i>;<br/>


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

<b>thus </b><a NAME="E10:21"/>
( <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_9"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:21_9"/>
<font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_9"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_9"/>
<font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></i>;<br/>


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

<b>thus </b><a NAME="E11:21"/>
( <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_10"/><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:21_10"/>
<font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_10"><i><font color="Green" title="E12">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_10"/>
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T30" title="VECTSP_4:th.30">VECTSP_4:30</a></i>;<br/>


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

<b>thus </b><a NAME="E12:21"/>
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_11"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:21_11"/>
( <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c11">M'</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:21_11"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_11"/>
<font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c9">n2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</a></i>;<br/>


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


</div>
