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<div><span class="kw">theorem </span><a NAME="T21"><span class="comment"><font color="firebrick">:: PENCIL_4:21</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">V</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">V</font> &amp; 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">V</font> holds <br/> <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">k</font>) is   <a href="pencil_1.html#NM2" title="PENCIL_1:NM.2">PLS</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#T17" target="_self" onmouseover="rs('pencil_4/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="pencil_4.html#T18" target="_self" onmouseover="rs('pencil_4/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="pencil_4.html#T19" target="_self" onmouseover="rs('pencil_4/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="pencil_4.html#T20" target="_self" onmouseover="rs('pencil_4/T20')" onmouseout="rh()">Th20</a>, <a class="ref" href="vectsp_9.html#T36" onmouseover="rs('vectsp_9/T36')" onmouseout="rh()">VECTSP_9:36</a></span>;<br/></div></div>
