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<b>let </b><font color="Maroon" title="c1">V</font> be  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="rusub_4.html#K2" title="RUSUB_4:func.2">Subspaces_of</a> <font color="Maroon" title="c1">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="E4:30"/><i><font color="Green" title="E21">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="c2">n</font> <a href="rusub_4.html#K2" title="RUSUB_4:func.2">Subspaces_of</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E2:30"><i><font color="Green" title="E20">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="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:30"/>
<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="E7:30"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="rusub_4.html#K1" title="RUSUB_4: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="c2">n</font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E4:30"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="rusub_4.html#D4" target="_self" title="RUSUB_4:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E8:30"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_4.html#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rusub_4.html#E7:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="rusub_4.html#T8" target="_self" title="RUSUB_4:th.8">Th8</a></i>;<br/>


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