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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">m</font>, <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <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="b1">V</font> holds <br/> not  <a href="pencil_4.html#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</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="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">m</font>, <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <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="Maroon" title="c2">V</font> holds <br/> not  <a href="pencil_4.html#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</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"> ( <font color="Maroon" title="c4">n</font> <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="Maroon" title="c2">V</font> implies  not  <a href="pencil_4.html#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>





<b>assume </b><a NAME="E1:27"/>
<font color="Maroon" title="c4">n</font> <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="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#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</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="c5">W</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <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="E3:27"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T39" title="VECTSP_9:th.39">VECTSP_9:39</a></i>;<br/>
<a NAME="E4:27"/>
<font color="Maroon" title="c3">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E3:27"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pencil_4.html#D6" target="_self" title="PENCIL_4:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
not  <a href="pencil_4.html#K6" title="PENCIL_4:func.6">SubspaceSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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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