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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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="b1">V</font> holds <br/> not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">k</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <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">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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="Maroon" title="c2">V</font> holds <br/> not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">k</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a></span><br/><b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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="Maroon" title="c2">V</font> implies  not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> )</span><br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E19">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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="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  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a></span><br/>

<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E2:21"/>
not the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="pencil_4.html#T11" target="_self" title="PENCIL_4:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:21"/>
not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</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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