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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<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="c2">K</font>;<br/>







<b>assume </b><a NAME="E1:20"/><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="E2:20"/>
( <font color="Maroon" title="c1">x</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="c1">x</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="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c1">x</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="c5">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#E1:20"><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="E3:20_1"/>
<font color="Maroon" title="c1">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_1"/>
<font color="Maroon" title="c1">x</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#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></i>;<br/>


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

<b>thus </b><a NAME="E3:20"/>
( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c4">N</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" title="E12">A3</font></i>: 
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">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#E1:20"><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="E3:20_2"/>
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c5">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_2"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_2"/>
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/>


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


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