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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><font color="Olive" title="b1">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b2">N</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <font color="Maroon" title="c2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">N</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:30"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">N</font></span><br/>

<a NAME="E2:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c2">M</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<a NAME="E3:30"/>
<font color="Maroon" title="c3">N</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_6.html#T4" target="_self" title="LMOD_6:th.4">Th4</a></span>;<br/>
<a NAME="E4:30"/><span class="kw">then </span>
<font color="Maroon" title="c2">M</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30"/>
<font color="Maroon" title="c2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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