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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:20_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/>


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

<b>thus </b><a NAME="E3:20"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:20_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T18">VECTSP_4:18</a></i>;<br/>


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


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