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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">N</font> holds <br/> <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b2">N</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b1">M</font></span><br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c3">N</font> implies  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">N</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c2">M</font> )</span><br/>





<b>assume </b><a NAME="E1:25"/>
<font color="Maroon" title="c2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">N</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">M'</font> = <font color="Maroon" title="c2">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:25"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">N</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">M'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T50" title="VECTSP_4:th.50">VECTSP_4:50</a></i>;<br/>
<b>hence </b><a NAME="E4:25"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">N</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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