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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="rlvect_5.html#V1" target="_self">finite-dimensional</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rlvect_5.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="rlvect_3.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_5.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:45"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E1:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rlvect_5.html#E3:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_5.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E38</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> =  <a href="rlvect_3.html#K1">Lin</a> <font color="Maroon">c<sub>5</sub></font> as  <a href="rlvect_1.html#V2">strict</a> <a href="rlvect_5.html#V1" target="_self">finite-dimensional</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E9:45"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E3:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_3.html#D3">RLVECT_3:def 3</a></i>;<br/>
<a NAME="E10:45"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T6">RLVECT_3:6</a></i>;<br/>
<a NAME="E11:45"/><b>then </b>
 <a href="rlvect_5.html#K1" target="_self">dim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E6:45"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_5.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E12:45"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="rlvect_5.html#K1" target="_self">dim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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