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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font> )
 ;<br/>

<a NAME="E2:33"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="pencil_4.html#K7" target="_self">GrassmannSpace</a> <font color="Maroon">V</font>,<font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D5">PENCIL_1:def 5</a>
<div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K7" target="_self">GrassmannSpace</a> <font color="Maroon">V</font>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K7" target="_self">GrassmannSpace</a> <font color="Maroon">V</font>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1"/>
not the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K7" target="_self">GrassmannSpace</a> <font color="Maroon">V</font>,<font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="txt" href="pencil_4.html#E4:33_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green">E32</font></i>: 
(  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E5:33_1"/>
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="pencil_4.html#E4:33_1"><i><font color="Green">E32</font></i></a>, , </i>;<br/><a NAME="E6:33_1"/><b>then </b>
 <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T31">VECTSP_9:31</a></i>;<br/><b>hence </b><a NAME="E7:33_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="vectsp_9.html#T31">VECTSP_9:31</a>, <a class="txt" href="pencil_4.html#E4:33_1"><i><font color="Green">E32</font></i></a></i>;<br/></div>
<b>end;</b></div>


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