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<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">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">GF</font>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">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="c3">V</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">I</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c4">I</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D1" target="_self" title="VECTSP_9:def.1">Def1</a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:42"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:42"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="vectsp_9.html#E3:42"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="vectsp_9.html#D2" target="_self" title="VECTSP_9:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">I</font><b> such that </b><br/><a NAME="E7:42"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E4:42"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="finseq_4.html#T87" title="FINSEQ_4:th.87">FINSEQ_4:87</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">W</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V</font> ;<br/>
<a NAME="E10:42"/>
<font color="Maroon" title="c4">I</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:42"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E11:42"/><b>then </b>
<font color="Maroon" title="c6">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</a></i>;<br/>
<a NAME="E12:42"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E7:42"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="vectsp_9.html#T30" target="_self" title="VECTSP_9:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E13:42"/>
 ex <font color="Olive" title="b1">W</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">V</font> st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/>


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