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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being <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>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Olive" title="b1">V</font> holds <br/> not <font color="Maroon" title="c1">n</font> <a href="rlvect_5.html#K2" title="RLVECT_5:func.2">Subspaces_of</a> <font color="Olive" title="b1">V</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c2">V</font> implies  not <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>



<b>assume </b><a NAME="E1:47"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">W</font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">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="E3:47"/><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="c3">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#E43"><i><font color="Green" title="E35">Lm2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:47"/>
not <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="rlvect_5.html#E3:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="rlvect_5.html#D4" target="_self" title="RLVECT_5:def.4">Def4</a></i>; <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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