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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>;<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>;<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>;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E22">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> )
 ;<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/>
<b>thus </b><a NAME="E2:24"/>
 <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#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a><br/>

<a NAME="E1:24_1"/>
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="txt" href="pencil_4.html#E1:24"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="pencil_4.html#T19" target="_self" title="PENCIL_4:th.19">Th19</a></i>;<br/>
<a NAME="E2:24_1"/><b>then </b>
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="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">W1</font>, <font color="Maroon" title="c7">W2</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c6">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c6">W1</font>,<font color="Maroon" title="c7">W2</font>,<font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D4" target="_self" title="PENCIL_4:def.4">Def4</a></i>;<br/>
<a NAME="E5:24_1"/>
not <font color="Maroon" title="c5">X</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E4:24_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="pencil_4.html#E1:24"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="pencil_4.html#T17" target="_self" title="PENCIL_4:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></i>;<br/>


</div><b>end;</b></div>


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