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<a NAME="E1:4_1_1"/>
 <a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">VS</font> is    <a href="vectsp_8.html#M1" target="_self">SubVS-Family</a> of <font color="Maroon">VS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_8.html#D2" target="_self">VECTSP_8:def 2</a><br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">VS</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">VS</font><b> such that </b><br/><a NAME="E3:4_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:4_1_1_1"/>
<font color="Maroon">x</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">VS</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_1_1"/>
 <a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">VS</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="vectsp_8.html#M1" target="_self">SubVS-Family</a> of <font color="Maroon">VS</font>
 ;<br/>


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