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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E37">A1</font></i>: 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="rlvect_5.html#E2:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">W</font> being  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:48"/>
<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>and </b><br/><a NAME="E7:48"/><i><font color="Green" title="E40">A4</font></i>: 
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="rlvect_5.html#E4:48"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="rlvect_5.html#D4" target="_self" title="RLVECT_5:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E8:48"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_5.html#E1:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="rlvect_5.html#E7:48"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="rlvect_5.html#T29" target="_self" title="RLVECT_5:th.29">Th29</a></i>;<br/>


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