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<a NAME="E1:4_1_1"/>
 <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">VS</font> is    <a href="vectsp_8.html#M1" title="VECTSP_8:mode.1">SubVS-Family</a> of <font color="Maroon" title="c2">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_8.html#D2" target="_self" title="VECTSP_8:def.2">VECTSP_8:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">VS</font> implies <font color="Maroon" title="c3">x</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font> )</span><br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">VS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">W</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E3:4_1_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</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="E4:4_1_1_1"/>
<font color="Maroon" title="c3">x</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E3:4_1_1_1"><i><font color="Green" title="E2">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_1_1"/>
 <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">VS</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="vectsp_8.html#M1" title="VECTSP_8:mode.1">SubVS-Family</a> of <font color="Maroon" title="c2">VS</font>
 ; <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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