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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lmod_6.html#NK1" title="LMOD_6:NK.1">Submodules</a> <font color="Maroon" title="c2">V</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">W</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:19_1_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/><b>thus </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon" title="c3">x</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:19_1_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1"/>
 <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font> is    <a href="lmod_7.html#M1" title="LMOD_7:mode.1">SUBMODULE_DOMAIN</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#D1" target="_self" title="LMOD_7:def.1">Def1</a></i>;<br/>


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