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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font>
 ;<br/>

<b>consider </b><font color="Maroon">I</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">I</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:42"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:42"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E7:42"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T87">FINSEQ_4:87</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font> as  <a href="vectsp_1.html#V11">strict</a> <a href="matrlin.html#V2">finite-dimensional</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E10:42"/>
<font color="Maroon">I</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E11:42"/><b>then </b>
<font color="Maroon">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T2">VECTSP_7:2</a></i>;<br/>
<a NAME="E12:42"/><b>then </b>
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:42"/>
 ex <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


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