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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> holds <br/> not <font color="Olive" title="b1">n</font> <a href="rusub_4.html#K2" title="RUSUB_4:func.2">Subspaces_of</a> <font color="Maroon" title="c1">V</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> implies  not <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> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:28"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <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> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<a NAME="E2:28"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st  <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rusub_4.html#E23"><span class="lab"><font color="Green" title="E16">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:28"/>
not <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> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#D4" target="_self" title="RUSUB_4:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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